A class of non-parametric tests in the competing risks model for comparing two samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D-1, obtained when the process is under control. Our main goal is to test if the failure rates in D-1 are equal to or less than the failure rates in another data set D-2, against "undesirable" neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests.